Problem: Milani's teacher draws students names at random, calls on the student, and replaces the name so that students know they should always be prepared to respond. There are $20$ students in Milani's class. Let $X$ be the number of names it takes for the teacher to draw Milani's name. Find the probability that the teacher first draws Milani's name as the $7^{\text{th}}$ name. You may round your answer to the nearest hundredth. $P(X=7)=$
Without a fancy calculator On each draw: $P({\text{Milani}})=\dfrac{1}{20}$ $P(\text{other student}})=\dfrac{19}{20}$ If the teacher first draws Milani's name on the $7^{\text{th}}$ try, then the teacher must first draw other student's names $6$ times, then draw Milani's name. Because the teacher replaces the names, each probability is independent, so we can multiply the probabilities of each event. $\begin{aligned} P(X=7)&=P(\text{OOOOOO}} {\text{M}}) \\\\ &=\left(\dfrac{19}{20}}\right)\left(\dfrac{19}{20}}\right)\cdots\left(\dfrac{19}{20}}\right)\left({\dfrac{1}{20}}\right) \\\\ &=\left(\dfrac{19}{20}\right)^6\left(\dfrac{1}{20}\right) \\\\ &\approx0.03675 \end{aligned}$ $P(X=7)\approx 0.03675\approx0.04$